This work is funded as an Engineering Initiation Award (EIA). The objective is to investigate the effects due to the helical beam electron trajectories in the electron cyclotron maser. Using a Maxwell- Vlasov theory and the telegraphist equations, the maser interaction from a cold tenuous annular electron beam in a lossy cylindrical waveguide (gyrotron amplifier) and a complex cavity (gyrotron oscillator) is to be analytically examined. Coupling among the beam, the higher order TE(mn) mode and space charge effects (to a first approximation, the TM(rs) mode) in the gyrotron amplifier is expected to be demonstrated at all frequencies, including the fundamental and harmonic cyclotron frequencies. The significance of the TM(rs) mode is to be examined. Growth rate and amplitude threshold dependence on the magnetostatic guide field, wall radius and wavenumber will also be examined. Mode coupling and competition will be analyzed in a gyrotron oscillator comparable to experiment. Field correctly satisfying boundary conditions will be used. From a total power analysis of the Beam-wave interaction, the significance of a parasitic TE mode or a space charge TM mode will be investigated. Understanding these coupling effects will provide more efficient, accurate and possibly more flexible gyrotron designs. The theory has potential for futuristic amplifier designs. This work involves theoretical investigations of the electron cyclotron maser (gyrotron) aimed at improving its performance as a amplifier for coherent radiation in the far infrared. The cyclotron maser has important potential applications in short-wavelength radar, communication links, spectroscopy, advanced accelerators and plasma heating in thermonuclear fusion reactors.